The comparison test for series of nonnegative terms

Identifier: F:creal-sumrange-comparisontest
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

forall a b, (forall k, le zero (a k)) -> (forall k, le (a k) (b k)) -> (Exists M, Converges (sumRange b) M) -> Exists L, Converges (sumRange a) L -- the comparison test as usually stated (pointwise 0 <= a k <= b k, sum b convergent), built by converting to sumRange_converges_of_dominated's Cauchy-hypothesis form: 0 <= a k and a k <= b k combine into abs (a k) <= b k via abs_le, and the witness M in the convergent-b hypothesis is eliminated into Cauchy (sumRange b) before that theorem is applied directly.

Formal statement

theorem CReal.sumRange_comparisonTest : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : ((x1 : AxNat) -> CReal)) -> ((x2 : ((x2 : AxNat) -> CReal.le CReal.zero (x0 x2))) -> ((x3 : ((x3 : AxNat) -> CReal.le (x0 x3) (x1 x3))) -> ((x4 : Exists.{1} CReal (fun (x4 : CReal) => CReal.Converges (CReal.sumRange x1) x4)) -> Exists.{1} CReal (fun (x5 : CReal) => CReal.Converges (CReal.sumRange x0) x5))))))

Dependencies

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12 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.sumRange_comparisonTest

Kind: kernel-term
Status: checked

Supports: CReal.sumRange_comparisonTest is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- sumRange_comparisonTest 2>/dev/null | grep -cE '^CReal\.sumRange_comparisonTest[[:space:]]'

Evidence notes

build_creal_prelude admits CReal.sumRange_comparisonTest through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command currently prints exactly one matching line for CReal.sumRange_comparisonTest. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.sumRange_comparisonTest

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the creal prelude's trusted surface is empty, which bounds CReal.sumRange_comparisonTest

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free creal

Evidence notes

Re-measured on this tree: creal: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: creal trusted surface = 0'. That bounds every declaration in the creal environment. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/series.rs, declare_sum_range_comparison_test)",
  "source": "theorem name and canonical type read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint(), theorem_dependencies()) -- no in-tree example prints this theorem's own type directly. Direct dependency edges cross-checked against theorem_dependency_inventory's own output on the same tree. The probe was built and run in the session scratchpad and deleted after use; crates/ source was not touched to produce this batch."
}